Formal orthogonal polynomials for an arbitrary moment matrix and Lanczos type methods∗
نویسندگان
چکیده
We give a framework for formal orthogonal polynomials with respect to an arbitrary moment matrix. When the moment matrix is Hankel, this simplifies to the classical framework. The relation with Padé approximation and with Krylov subspace methods is given. 1 Formal block orthogonal polynomials We consider a linear functional defined on the space of polynomials in two variables, defined by the moments μij = μ(wizj), i, j ∈ N. Let M = [μij ] be the (infinite) moment matrix, then for two polynomials p(w) = wp and q(z) = zq (w = [1, w, w2, . . .] and z = [1, z, z2, . . .], p,q ∈ C∞×1), we define a formal inner product by 〈p, q〉 = μ (p∗(w)q(z)) = p∗Mq. We call g(ω, ζ) = μ (1/[(ω − w)(ζ − z)]) the generator for the matrix M , since at least formally:
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